All---Or Nothing at All

All or nothing at all.Half a love never appealed to me. . . .---Words by Jack Lawrence, music by Arthur Altman

The appeal of mathematics to novelists and playwrights, while rare, may be increasing---I have in mind Tom Stoppard's play Arcadia, which is full of math (Fermat, chaos). Apostolos Doxiadis, a colorful figure on the Athens scene, who earned a master's degree in applied mathematics and is now a stage and screen writer and director, has given us an additional such production. His novel Uncle Petros dramatizes the sad life of a mathematician who considers himself a failure because he hasn't been able to achieve what hundreds of his predecessors were also unable to achieve.

Uncle Petros introduces the reader to a young mathematician, Petros Papachristos, born in 1895, who as a child was good at math and whose teachers told him he was a genius. Somewhat later, Petros got hooked on solving the Goldbach conjecture. He eventually gave up the attempt and spent the rest of an isolated life gardening and playing chess. Are there echoes here of artist Marcel Duchamp, who created the Nude Descending a Staircase and other controversial artworks, raised much éclat, and then retired to a life of chess? Judging from a plug by the well-known neurologist Oliver Sacks, author of The Man Who Mistook His Wife for a Hat, I infer that Doxiadis intends us to regard the action of Petros as irrational in the highest degree-another chapter in the chronicles of strange mental aberrations.

The book is targeted for a general readership and therefore makes few mathematical demands on non-mathematical readers. While Uncle Petros is no great piece of literature as such---the prose strikes me as rather wooden---readers will undoubtedly be pulled in by the story of a man who abandoned family, love, friendship, wealth, and position in a futile hunt for the right sequence of arcane symbols that would be acknowledged by peers as constituting a proof. Captain Ahab and his mad chase after Moby Dick come to mind.

Professionals, who know a bit of the Goldbach story and the mathematics scene, will find nothing new in the book either mathematically or psychologically. Our profession has lots of in-jokes that point to mild forms of Petros's kind of monomania as a not infrequent way of mathematical life.

What is Goldbach's conjecture? Christian Goldbach, an amateur mathematician and a correspondent of Euler, conjectured in 1742 that every even number greater than two is the sum of two primes. You can go on the Web and in a few minutes find a site that, for modest values of 2n, will provide you with all such prime decompositions. You will also find buckets of information on the many labors toward a proof of the conjecture, achievements for which it can be said "Closer, and closer, but still no cigar."

Doxiadis's fiction is partly "faction." I've heard that the character of Petros was inspired by the Princeton career of Christos Papakyriakopoulos, who got hung up on the Poincaré conjecture. Doxiadis weaves many historic figures into his plot, portraying both their mathematical personas and their psychosocial abnormalities. Wisely, he does not mention the "closer and closer" achievements or their associated technicalities. On the other hand, Gödel and Turing and their incompleteness theorems are brought in to provide Uncle Petros with an excuse for his failure. The Goldbach conjecture may just be one of those statements that are impossible of proof.

The dénouement is both tense and downbeat; it is one that I feel sure will reconfirm lay readers' frequent opinion or suspicion that mathematics is impenetrable and that its cultivators are all mad.

*****

In the course of describing Petros's mania, Doxiadis puts these words into his mouth:

"Listen to me: the way I see things in mathematics as in the arts---or in sports, for that matter---if you're not the best, you're nothing. A civil engineer, or a lawyer or a dentist who is merely capable may yet lead a creative and fulfilling professional life. However, a mathematician who is just average---I'm talking about a researcher, of course, not a high-school teacher---is a living, walking tragedy."

I would like to devote the rest of my space to a discussion of this sentiment. It's a sentiment that is common enough in the mathematical world. It is fostered mostly, but not exclusively, by pure mathematicians in their teaching, in their personal contacts with students and colleagues. It is reflected in their attitudes toward places of employment, specialties, individual accomplishments, prize awards, and displayed honorifics (e.g., "The Fergus F. and Mollie J. Ferguson Professor of Creative Concatenations").

G.H. Hardy had it bad. He used to rank mathematicians in linear pecking order, comparing them with star cricketers. By contagion, he passed his views on to such eminences as Mary Cartwright. My late Brown colleague Otto Neugebauer had it bad; he used to reduce person after person, on both sides of the Atlantic, to zeros. Another mutual colleague, in reaction and in disapproval, said of Neugebauer, not entirely fairly, "You see, he has a true-false, zero-one mind. The typical strength and debility of mathematicians."

I think it was Jean Dieudonné who said that mathematics needs the work of only about one hundred people. One can safely throw out everything else. The most eloquent and intellectualized discussion of this sentiment occurs, to my knowledge, in a long article in The New Yorker (February 19, 1972) entitled "Mathematics and Creativity," written by Alfred Adler. (The Mathematical Experience, by Reuben Hersh and this reviewer, contests Adler's brutally and apocalyptically romantic view; see pages 60-65.) In view of the publication of innumerable tedious and idiotic lists of "the hundred best whatnots" that the millennium has spawned, in view also of the solutions in recent years of several famous unsolved mathematical problems, in which the publicity tends to stress the individual who delivered the coup de grace, I would like to vent once again my intense dislike of this lopsided attitude toward mathematical creativity.

It is abundantly clear that genius exists or, more specifically, that a wide distribution of mathematical talent exists. It seems to be the case that great talent is in part inborn but that it can be augmented by education, opportunity, luck, and what I call the "royal jelly" of constant familial praise and peer acclaim. Genius, indeed creativity at any level, often manifests itself unconsciously in that it does not come directly when called upon.

It is also true that mathematics at the research level is, and has always been, highly competitive. It is very likely that the discipline will remain competitive. To some, there is no better ego food, no better high, than to do something that others, for all their trying, have not been able to do. Competition is an element that serves to stimulate genius, but more importantly, to elicit new ideas and possibilities.

But think of this: We are none of us born into a world we made. And mathematicians are born into a pre-existing world consisting of millions of words, symbols, and ideas of which the best of us can learn or know only a fingertip's worth, and to which we are invited to contribute to the best of our abilities.

The top levels of thought do not exist disconnected from this inheritance. They are firmly lodged within a solid and slowly constructed base of extensions, generalizations, explications, simplifications, new problems, applications, oral traditions, and metaphysical stances, along with the value judgments assigned by each era to this vast accumulation of material. This corpus, to my mind, is the proper way of regarding the giant that appears in the remark, often ascribed to Newton, that if he has seen farther, it is that he was standing on the shoulders of giants. Leonardo da Vinci---or possibly Icarus---may have invented the airplane, but a good part of the technological revolution had to occur before the emergence of a model that worked.

I could go on by advocating that more attention be paid to the claims of social constructivism as a proper philosophy for the field of mathematics. I could question the permanence or the ultimate value of today's acclaimed accomplishments. What fraction of our professionals have ever examined or used the great achievement of Lindemann, who proved in 1882 that pi is not an algebraic number and thus put to rest a question that had been around since classical antiquity? What fraction of a solid text on, say, matrix theory is the fruit of neurotic fanaticism of the sort described by Doxiadis? For all we know, the future may identify as the key mathematical achievement of the 20th century the construction of Fortran, which, at the moment of its birth, I and a host of colleagues dismissed as merely work for mathematically trained schlepps.

I want to end by quoting (with approval) the balanced judgment of the American psychologist and philosopher William James:

"The community stagnates without the impulse of the individual; the impulse dies away without the sympathy of the community."